Fractals and Time, Part II: Spacetime Smearing and Crystalline Time

by Christopher Vitale

What follows is Part II of series of posts on the relation between fractals and time. For the first post in this series, see here. [Note: Parts II, III, and IV were written separately from Part I, so there may be some overlap].

Before we get to the specifics of how time might be considered a fractal, we need to get a specific sense of how quantum phenomenon relate to time.

Smearing. In the time after a quantum particle vanishes from view and becomes a potential particle, the range in which this particle may appear expands, based on the speed of this particle, which is the speed of light for photons, slower for heavier particles. This range may be thought of as a sphere (which may be distorted by curvature in spacetime) which expands in size over time until that potential is localized by means of an interaction. If a test particle is shot into the spacetime area in which a potential is located, there are some sub-areas in which it is more likely that the original particle will emerge and actualize in relation to the test particle. Much of this is determined by the trajectory of that particle in relation to the event in which it last appeared. We cannot be sure that the particle which emerges on either side of a potential is the same particle, and in fact, if all potentials might not be in some sort of commuication. We do know, however, that the previous interaction impacts the forthcoming one. Furthermore, as quantum eraser experiments have now proven, fortcoming events also impact the ways in which a potential may actualize, for quantum potentials interfere with themselves differently over multiple trials depending on events which occur after both interactions on either side of a quantum potential have occurred. Quantum particles act as if causality not only flowed forward in time, from cause to effect, but also effect to cause. This is why they may be described to exist within a reversible sort of time, for there is no way to know if the particles are actually going backwards or forwards in time, particularly because the only way to distinguish quantum particles going forward in time from those going backwards would be a shift in spin.

In this sense, it is meaningless to say that quantum potentials exist in time in the manner that quantum particles do. Likewise, because quantum particles may actualize anywhere within a given spatial location, if in differing probabilities, we cannot say that these potentials are localized in any particular place within that given area. It is in this sense that we say that in a given spacetime area, determined by a spheroid shape and in relation to external measures of spacetime, that for a given quantum potential within that spacetime area and between quantum events, that spacetime in the traditional sense does not exist, but rather, is smeared, and that, furthermore, it is as if the particles themselves were, as in a sense, smeared across the spacetime area in question.

What is it that determines the probability to which a quantum potential might actualize within its relevant spacetime area? If we are dealing with a static particle detector, the more direct a path between the emission of a potential and the detector, the more likely the particle will actualize there. Quantum randomness, either caused by microinfluences from the potential’s context or some other unknown source, has manifested in these experiments such that over multiple trials, there is only probability, not certainty, as to where that particle will land. If we shoot a test particle into a spacetime area relevant to a quantum potential, there is a higher probability of the two potentials actualizing as an event on a path which is the most direct between the two potentials, as based on their trajectories when they were emitted.

For the reasons listed above, many have suggested that a quantum potential ‘explores’ all paths within its spacetime area, but explores with more strength those which are more direct. Were quantum potentials to exist in time, it would be as if the quantum potential would run through all paths, forewards and backwards in time at the same time, each path taking the same amount of time to traverse, and such that the potential would run through the most direct paths more frequently, making it more likely that if disturbed it would show up in the direct paths rather than others. If we can imagine this to occur, but not progressively in time but statically at the same time, we have a sense in which it seems quantum potentials exist.

Crystal. Such a state, while static and outside of time, nevertheless changes, and here we will employ the term used by Whitehead, ‘advance’, for this sort of development which occurs in states which seem to be outside of standard forms of time. Quantum potentials advance because the size of the spacetime area in which a particle may actualize increases over time. In this sense, the smeared area of spacetime, or the smearing of the particle as potential in spacetime, increases over time as percieved outside the potential in question. As a spacetime potential expands as it advances, each new added area, and the time associated with it, recalibrates the entire set of probabilities within the quantum potential, both forwards and backwards in time, and if we think of separate paths off the most direct ones as sideways, then we can say in many directions in spacetime at once.

This is why some researchers have referred to this sort of time as spatial, while others have put forth a fractal model to describe such phenomenon, and both describe aspect of what is at work in quantum potentials. We will describe this sort of spacetime, following Gilles Deleuze, as crystalline. For like a crystal, a quantum potential grows from a germ, namely, a quantum event which emits a potential. The potential then grows, in all spacetime directions, in a manner which is both determined by that germ and the medium, or context, within which that germ finds itself, as well as some degree of randomness.

As exponents of the fractal metaphor have argued, it seems that quantum potentials exhibit fractal properties at multiple levels of scale, similar to the manner in which cells of a crystal repeat at multiple levels of scale. As with crystals, as quantum potentials advance, they increase in spacetime area in a manner in which each new increment is mediated by the shape of the crystal as a whole, and this is due to the fractal iterative structure at work in both part and whole. As with holographs, the whole is represented, if in mediated form, within each of the parts, in a manner similar to the iterative nature of crystals. And just as light is refracted when it enters a crystal according to the manner in which the whole is enfolded in its parts, so it is that the whole of a quantum potential is enfolded in all the parts (otherwise it could not advance at its edges), if differently and more intensely at some points than others, thereby leading to the refraction of probability states in a manner analogous to that of light. It is in this manner that quantum potentials advance in a crystalline manner, even if they do so in a manner which exceeds traditional definitions of space and time.